On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain

Abstract

We study two seminal approaches, developed by B. Simon and J. Kisyński, to the well-posedness of the Schrödinger equation with a time-dependent Hamiltonian. In both cases the Hamiltonian is assumed to be semibounded from below and to have constant form domain but a possibly non-constant operator domain. The problem is addressed in the abstract setting, without assuming any specific functional expression for the Hamiltonian. The connection between the two approaches is the relation between sesquilinear forms and the bounded linear operators representing them. We provide a characterization of continuity and differentiability properties of form-valued and operator-valued functions which enables an extensive comparison between the two approaches and their technical assumptions.

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Cited by in corpus (5)

• On global approximate controllability of a quantum particle in a box by moving walls
• Quantum controllability on graph-like manifolds through magnetic potentials and boundary conditions
• On the applicability of Kolmogorov's theory of probability to the description of quantum phenomena. Part I: foundations
• On a sharper bound on the stability of non-autonomous Schrödinger equations and applications to quantum control
• Global approximate controllability of quantum systems by form perturbations and applications